Happy new year everyone! I expect to get back to posting every two weeks, and I am going to start to catch up with the backlog of comments to respond to.
The ninth post is the January 8 version here, ending with 14.1. Section 13.4 on valuative criteria is included, but I haven’t edited it yet in response to the useful comments here, and hope to do so by the next posting.
For learners.
Read 12.3, 13.1, 13.2, and 13.3. Skip 13.4 (because I haven’t edited it). Likely skip starred sections 12.4 and 13.5 (depending on your taste), and feel free to read ahead to 14.1.
12.3 is on a couple of very useful codimension one theorems (Krull and Hartogs). It is most important to get comfortable with their statements and applications. The proofs are less essential.
Starred section 12.4 proves Krull’s theorem, and introduces Artinian rings.
13.1 is intended to define the Zariski tangent space, and to show you that it is easy to work with.
13.2 defines the algebro-geometric version of smoothness.
13.3 is on discrete valuation rings (“smoothness in codimension 1”), which prove to be of critical importance.
You should skip 13.4 on valuative criteria (because I haven’t edited it), and likely also the starred section 13.5, on completions (power series).
14.1 is an introduction to vector bundles and locally free sheaves. There isn’t much content, but there is an important and subtle change in perspective from what you may be used to. That’s why I wanted you to take a look at it.
The ten exercises you should most do:
12.3.C, 12.3.E, 13.1.B, 13.1.E, 13.1.G, 13.2.B, 13.2.E, 13.2.J, 13.3.A, 13.3.G.
Exercises involving explicit examples that you should sample: 12.3.B, 13.1.C, 13.1.D, 13.1.H, 13.2.C, 13.2.F, 13.2.H, 13.2.K, 13.3.B.
If someone tries 12.3.G (successfully or unsuccessfully), please tell me. I want this exercise to be do-able, and I’ve added a hint that I hope might help. (An earlier version was too hard.)
You may find it worthwhile to read section 12.4 on the proof of Krull’s Principal Ideal Theorem, because it will introduce you to Artinian rings. If you try and find something confusing, please let me know.
For experts.
The (optional) proof of Krull gives an opportunity to introduce Artinian rings and modules, which frankly don’t enter the story later in the notes (despite of course being important “later on” in much of the field). So this is a good place to let the most interested of readers know the main facts, which they should be able to prove themselves. Question: What the facts about Artinian rings people should most know? The ones I can think of (that aren’t currently there) are the following. Artinian rings are Noetherian. Artinian rings over a field are finite type. A finite type k-ring is Artinian iff it is dimension 0. Anything else? (And is there a proof of Krull that you like better?)
My sense is that when you learn algebraic geometry, and you know that you like the idea of smoothness, you want A^n_k to be smooth. But it is a hard fact that this is true (for n at least 3). My current position is that one really doesn’t care about regularity in general (at this stage in learning, for most people). Regularity in dimension 1 is of course important.
What is the right definition of node (of a curve), cusp, or tacnode? Currently I give the definition over an algebraically closed field: it should be a point of a finite type scheme, of dimension 1, whose completion looks as you expect. This seems a bit clunky, but I can’t think of anything better. I’ve chosen to duck the question over a general field, but perhaps I shouldn’t. (I would define it by base changing to the algebraic closure. But I’d prefer to avoid issues such as the fact that the point can split into several points, and then one is a node if the others are. Readers interested in such things should probably think through this themselves. Readers less interested will be distracted — and many complex-minded readers will be very interested in the notion of node, cusp, etc., and I don’t want to make it seem harder than it is for them.)
Math 216: Foundations of Algebraic Geometry 
January 8, 2011 at 3:42 pm
Ravi, 14.1.9 makes me wonder if it is reasonable to suggest that quasi-coherence is a good concept beyond the scheme setting. Or rather, since anyone can make up whatever definition they want, it should be warned somewhere that this notion can have some unexpected properties when adapted beyond the scheme case. (I give some examples of Gabber over the analytic unit disc in my paper “Relative ampleness in rigid geometry”.)
As for defining some singularity types in terms of completed local ring on a geometry fiber, I think one needs Artin approximation to prove that such definitions are equivalent to other formulations which are more geometrically appealing (e.g., for a node, equivalent to an etale-local criterion).
Above you mention the issue of smoothness of affine space; I think you meant “regularity” (since smoothness of affine space is quite easy…according to one definition of smoothness, though equivalence with others requires real work).
February 8, 2011 at 5:21 am
I agree with your first paragraph, but don’t have much experience outside the scheme setting.
About definition of singularity types: I agree that one needs Artin approximation to link etale-local and formal-local definitions. I wonder whether someone seeing the subject for the first time (and who might not see it for a second time if they are from a nearby field) might find the power series definition more “geometrically appealing”, in that they don’t have to wait too many hundreds of pages before knowing what a node is.
And indeed I meant regularity, thanks!
January 12, 2011 at 3:07 am
Concerning an older part, but this is still unfixed in the latest notes:
6.5.1: In (1) you probably want to say “The generic points of the irreducible components of a locally Noetherian [scheme are associated points]”, but you forgot what I have put into [].
January 13, 2011 at 3:11 pm
7.4.A: Is there a reason why this would not work for d=0?
February 8, 2011 at 5:25 am
For 6.5.1: thanks, fixed! And for 7.4.A: d=0 won’t work because we need elements of S_+ to map to elements of R_+. (If I should say more, just let me know.)
February 8, 2011 at 3:18 pm
You are right of course. I wanted to use it for 9.2.Q at that time, but of course it does not make sense. Do you have a hint for 9.2.Q?
February 9, 2011 at 8:59 am
Suppose f \in S_+. I will describe a map from D(f)=Spec (S_*)_f in Spec S_* to Proj S_*. In fact, it will map to the open set of S_* that is Spec ((S_*)_f)_0 (where as usual those 3 subsets mean 3 different things). To do this, I need to describe a ring ma ((S_*)_f)_0 –> (S_*)_f. I’ll just take the inclusion (and this map feels very natural!). So what is left is to check that all of these maps glue together. Please let me know if I should say more!
January 12, 2011 at 8:57 am
I’m also going back in time with this question. In 12.2.H what does X have to do with V?
January 25, 2011 at 9:45 am
Whoops! X and V are supposed to be the same. Now fixed. Thanks!
January 13, 2011 at 4:19 pm
9.2.D: Do you have a hint for this? It seems hard to me.
January 13, 2011 at 4:25 pm
It seems that Hartshorne does this in II, Corollary 5.16 (on page 119), but he invokes things that I do not understand yet.
March 25, 2012 at 5:23 pm
This is a long-delayed response, but hopefully it will be of help to later readers. Here is a hint. The alleged graded ring is described, as corresponding to a graded module. Check that this works on a projective distinguished affine. (I have solutions people handed in for the 2011-12 course, so I can send anyone a working solution.)
January 14, 2011 at 5:55 am
I had trouble with 12.2.D. Then I looked into Eisenbud’s Commutative Algebra and the Step I am missing seems to be Corrolary 13.4 (dim I + codim I = dim R). However he uses an expanded version of your theorem 12.2.1 (his theorem A on the same page).
It includes: “every maximal chain of primes in R has length dim R”.
But he uses an extended version of the Noether normalization theorem to prove that.
Do you know a way to do the Exercise without using all of that?
January 22, 2011 at 6:43 am
The fact dim I + codim I = dim R, where R is an affine domain, can be proved without extended Noether’s normalization as follows (We will use only Theorem 12.2.1 and properties of integral extensions).
First of all, it is enough to prove the assertion for a prime ideal I=P. There are two steps in the proof.
Step 1. I claim that, If P is a minimal prime nonzero ideal, then 1 + dim R/P = dim R.
Proof: If R is a polynomial ring in n indeterminates, then R is a UFD. Thus P=(f) for some irreducible polynomial. Thus, dim R/(f)=n-1 (Here we use Theorem 12.2.1, that is n = dim R).
If R is an arbitrary integral domain, then, using Noether’s normalization, we have A\subset R, where A is a polynomial ring and R is integral over A. If P’=A\cap P, then P’ is minimal nonzero ideal in A (since A is integrally closed and R is a domain, we can use the going-down property). Thus, 1 + dim A/P’ = dim A. But dim A/P’ = dim B/P and dim A = dim B (because they are integral extensions).
Step 2. For any saturated chain of primes P_k\supset … P_1\supset P_0=0 in R, we have 1 + dim R/P_1 = dim R. Then 2 + dim R/P_2 = dim R. At the end, we get k + dim R/P_k = dim R. But P_k is maximal, so, k = dim R. From that our assertion immediately follows.
I could not find the going-down theorem in the Ravi’s course. The easiest proof I know can be found in Atiyah Mcdonald “Introduction to commutative algebra” Theorem 5.16 p. 64. Another proof based on Galois theory is contained in Matsumura “Commutative algebra 2ed”(5.E) Theorem 5(v) p. 33. The latter proof is more impressive but harder.
August 23, 2011 at 10:12 am
D.H., I agree with you — the problem is too hard. A proposed exposition, which follows Dima Trushin’s argument, is here.
To Dima: thanks! I like this approach better than Mumford’s (given what readers will know at this point).
March 25, 2012 at 5:25 pm
Update: this has been turned from an exercise to a theorem (12.2.9). This was done some time ago.
January 14, 2011 at 8:14 am
Hi,
I was just reading over the notes again and noticed that there is a typo on
Page 85, Example 3, 2nd line:
the symbol for affine line.
January 16, 2011 at 8:40 am
Hi,
Also on page 111 you reference page 102 Exercise 4.6.T but it should refernce Exercise 4.6.S.
January 16, 2011 at 9:58 am
I should have probably submitted these as a list but oh well,
On page 77 Exercise 3.7.D line 3
The index on the triple overlap of isomorphism is wrong.
January 25, 2011 at 9:44 am
Thanks for these! I’ve fixed the third (about the triple overlap). I didn’t understand the previous two, but have some guesses why.
About p. 85: it looks right to me. Perhaps you have an earlier version and it is now fixed. Or perhaps it just looks imperfect: it should be a blackboard bold A, superscript 1, subscript k-bar, and the bar looks weird. Or perhaps I’m missing something.
And on p. 111, I think it should indeed be 4.6.T (which suggests the true fact that you can make disjoint unions of a finite number of Spec’s by taking Spec of the product). But if the notes you are looking at have parts with different dates, the exercise names may have changed.
So please let me know if I am missing anything!
February 1, 2011 at 8:56 am
SI feel a bit silly now.
I have different version of the notes printed it out, so in this outdated version 4.6.T is 4.6.S(the last exercise given is 4.6.S) hence my confusion.
On page 85: looking at it, it makes sense. I think it just looked a bit weird with the bar.
Thanks. Sorry for getting the mistakes wrong
April 24, 2011 at 11:23 am
No worries; thanks very much for the comments.
January 14, 2011 at 2:00 pm
13.1.B: Does this work for infinite dimensional tangent spaces? If so, how? Or should we assume that A is noetherian? Will we be using infinite dimensional tangent spaces anywhere?
February 8, 2011 at 5:11 am
It indeed works for infinite dimensional tangent spaces, with the usual convention that “infinity – 1 = infinity”. More precisely: the algebra fact will be true without any hypotheses on the ring. And m/m^2 is certainly a vector space over the field A/m. So the Zariski tangent space of A/(f) is a quotient of the Zariski tangent space of A by one equation, which may be zero. The dimension statement follows. Please let me know if I should say more!
January 16, 2011 at 3:35 am
10.3.H: Does this also work for schemes that are only _locally_ Noetherian?
February 8, 2011 at 5:16 am
Can you please clarify? There isn’t a 10.3.H in the current version, or in some of the previous versions.
February 8, 2011 at 2:51 pm
Sorry, I meant 13.3.H.
April 24, 2011 at 11:24 am
Indeed it does — and I’ve added this (in the version to be posted after Apr 24).
January 21, 2011 at 9:14 am
A member of the UChicago reading group (who prefers to remain anonymous) pointed out what appears to be a subtle omission in 12.2.10 (i.e., proof of 12.2.1), together with a simplification that avoids the issue: How do you know that p_1 contains a prime of height one (which we then assume wlog = p_1)? We have not shown that primes in k[x_1,…,x_n] necessarily have finite height.
It is easy to show that p_1 contains a height-one prime once we have the Principal Ideal Theorem, but it is unnecessary: regardless of the height of p_1, modding out by it introduces some relation among the x_i, and hence reduces the transcendence degree, so we can apply strong induction.
August 23, 2011 at 9:45 am
(Long delay in reply — sorry as always!)
Thanks, I’ve patched this, following your advice, but phrasing it differently. Instead, I say: let g be any nonzero element of p_1, and let f be any irreducible factor of g. Then replace p_1 by f.
January 21, 2011 at 3:13 pm
Proposition 12.3.5 holds for arbitrary integral domains (not only for Noetherian ones). The general proof can be found in I. Kaplansky “Commutative Rings” Theorems 3-5 pp. 3-4. Moreover, in the general case, the proof is much easier.
January 25, 2011 at 9:34 am
Thanks! I will take a look when I have a chance. I’ve found this online:
Update March 25 2012: I’ve unstarred this, but haven’t forgotten it; I have a flag in the file to follow up on this at some point.
Update December 10, 2012: The notes are now updated to reflect this. Thanks again to Dima Trushin!
April 15, 2012 at 4:37 pm
Does anyone know of a reference for why a smooth variety over a nonperfect field is nonsingular? (I’ve seen at least one reference in the literature that was not complete…) I know an argument (thanks to Brian Conrad), but surely this is something that someone has written down somewhere in a readable manner.
August 30, 2012 at 11:13 am
Answer: as one would hope, the answer is in the stacks project, see tag 00TT. I’ll add this to the notes (in 13.2.6) shortly.